On the positivity and integrality of coefficients of mirror maps

We present natural conjectural generalizations of the positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano toric varieties as a special case. We find that, given the combinatorial data from which one constructs a mirror pair of Calabi--Yau complete intersections, there are two ways of writing down an associated mirror map’: one which is the true mirror map', meaning the one which appears in mirror symmetry theorems; and one which is the naive mirror map’. The two are equal under a certain combinatorial criterion which holds e.g. for the quintic threefold, but not in general. We conjecture (based on substantial computer checks, together with proofs under extra hypotheses) that the naive mirror map always has positive integer coefficients, while the true mirror map always has integer (but not necessarily positive) coefficients. Most previous works on the integrality of mirror maps concern the naive mirror map, and in particular, only apply to the true mirror map under the combinatorial criterion mentioned above.

💡 Research Summary

The paper investigates the arithmetic properties of mirror maps associated with Calabi–Yau complete intersections constructed via the Batyrev–Borisov mirror symmetry framework. After reviewing the classic quintic threefold example—where the mirror map’s coefficients are famously positive integers—the authors generalize to the multi‑parameter setting of toric complete intersections. They introduce a combinatorial datum consisting of integer vectors (v_{ij}) and define two distinct mirror maps: a “naive” mirror map (\psi^{\mathrm{n}}{ij} = \exp(\varphi{ij}/\varphi_0)) and a “true” mirror map (\psi^{\mathrm{t}}{ij} = \exp\big((\varphi{ij}+\tau_{ij})/\varphi_0\big)). Here (\varphi_0) and (\varphi_{ij}) are hypergeometric‑type series built from the monoid (K_0) (the non‑negative kernel of the linear map defined by the (v_{ij})), while (\tau_{ij}) involves an auxiliary monoid (K_{ij}) that captures contributions from negative components in the kernel.

The authors impose a “Fano” condition on the data: the convex hull (\Delta) of the vectors together with the origin must be a lattice polytope whose only interior lattice point is the origin. Under this hypothesis (and the mild Assumption 1.1 that the origin lies in the interior of (\Delta) and the vectors span (\mathbb Z^d)), they formulate two conjectures. Conjecture A asserts that for any Fano data, each naive mirror map (\psi^{\mathrm{n}}{ij}) has integer coefficients, and moreover its logarithm (\varphi{ij}/\varphi_0) has non‑negative coefficients, i.e. the naive mirror map lies in the semiring (\mathbb N